Calculus for AP Physics

Introduction:

AP Physics C is a calculus-based physics course, but the fact is
that you can do pretty well, grade-wise, in AP Physics C -
particularly the mechanics part - without any knowledge of calculus.
Of course it won't hurt you to know what is going on with all those
strange mathematical symbols and concepts, and there will definitely
be problems on the AP Test that require a knowledge of calculus.
Fortunately, you can get by really well - even on the AP Test - with
very rudimentary calculus skills that can be learned in a few
minutes.

You need to know something about:

Derivatives - a derivative is a
rate of change, or graphically, the slope of the tangent line to a
graph. Although physics is "chock full" of applications of the
derivative, you need to be able to calculate only very simple
derivatives in this course.

Definite Integrals - a definite
integral represents an area, and there are a myriad of
applications of integrals in physics. In this course, you need to
be able to evaluate only very elementary integrals.

Derivatives:

A derivative is a rate of change, which, geometrically, is the
slope of a graph. In physics, velocity is the rate of change of
position, so mathematically velocity is the derivative of position.
Acceleration is the rate of change of velocity, so acceleration is
the derivative of velocity. Net force is the rate of change of
momentum, so the derivative of an object's momentum tells you the net
force on the object. These are only a few of the applications of the
derivative in physics.

Derivative Rules:

Finding the derivative of a function ("differentiating" in
calculus language) is a rule-based operation. In other words, you
need to recognize what derivative rule applies, and then apply it. In
order to recognize what derivative rule applies, you need to know
some derivative rules. The tables below list the derivative rules
that you will use in this course, and shows some examples of their
use. These rules are stated using "t" as a variable (the derivative
is "with respect to" t, in calculus language), since most of the
functions that we will use are functions of time. If you are taking a
derivative whose variable is "s," simply substitute "x" for "t" in
the derivative rule.

Rules for Particular Functions:

Rule in English

Rule in Math. Notation

Example

The derivative of a constant is zero.

If x(t) = 5, then v(t) = 0.

If v(t) = -3, then a(t) = 0.

The derivative of t is one.

If x(t) = t, then v(t) = 1.

The derivative of t to a power is the power times t to
the "one less" power.

If x(t) = t2, then v(t) = 2t1 =
2t. (n = 2)

If v(t) = t4, then a(t) = 4t3.
(n = 4)

If x(t) = t-3, then v(t) =
-3t-4. (n = -3)

The derivative of the sine of t is the cosine of
t.

If x(t) = sin t, then v(t) = cos t.

The derivative of the cosine of t is the negative of
the sine of t.

If v(t) = cos t, then a(t) = -sin t.

Rules for Combinations of Functions: In the rules below, u and
w represent functions of time, t.

Rule in English

Rule in Math. Notation

Example

The derivative of a constant times a function equals
the constant times the derivative of the function.

If x(t) = 3t2, then v(t) =
3(2t1) = 6t. (c = 3 and u = t2)

If v(t) = 4sin t, then a(t) = 4cos t. (c = 4, u = sin
t)

The derivative of the sum (or difference)of two
functions is the sum (or difference) of their
derivatives.

If x(t) = t + sin t, then v(t) = 1 + cos t. (u = t, w
= sin t)

If v(t) = t2 - 4t, then a(t) =
2t1 - 4(1) = 2t - 4. (u = t2, w =
4t)

The derivative of a composite (one function inside
another function) function equals the derivative of the
"outside" function leaving the "inside" function alone,
times the derivative of the "inside" function. (The Chain
Rule)

Definite Integrals:

A definite integral represents an area, and evaluating a definite
integral ("integrating" in calculus language) is the inverse of
finding a derivative - like subtraction is the inverse of addition.
In physics, the area under a velocity vs. time graph represents
displacement, so the definite integral of velocity gives
displacement. The area under an acceleration vs. time graph equals
change in velocity, so the definite integral of acceleration tells
you the change in velocity. The area under a force vs. position graph
equals work done by the force, so the definite integral of force
(with respect to position) tells you the work done by the force.
There are many, many more applications of the definite integral in
physics.

The diagram above shows the relationship between the
definite integral notation and the area it represents. "a"
and "b" - called "limits of integration" go at the bottom
and top of the big "S". "f(t)" is the function being
integrated (the "integrand"), and "dt" says that "t" is the
variable being used.

The notation is read "the definite integral from a to b
of f of t, dt".

Rules for Definite Integrals:

Just like you can subtract 5 from 12 by thinking, "What do I have
to add to 5 to get 12?", you can evaluate definite integrals by
thinking "What function would I have to differentiate to get the
function in this integral?" That function is called the "integral"
or "antiderivative." The mathematical symbol for the antiderivative
looks like the definite integral without the limits of
integration.

Antiderivative Rules for Particular Functions: If you are a
calculus student, you will notice that we are ignoring an
important mathematical point in the following rules.

Rule in English

Rule in Math. Notation

Example

The antiderivative of a constant is the constant times
t.

The antiderivative of t to a power is t to the
power-plus-one, divided by the power plus one.

(This is the inverse of the derivative formula )

The antiderivative of the sine of t is the negative of
cosine t.

(This is the inverse of the derivative formula .)

The antiderivative of the cosine of t is the sine of
t.

(This is the inverse of the derivative formula .)

Antiderivative Rules for Combinations of Functions: In the
rules below, u and w represent functions of time,
t.

Rule in English

Rule in Math. Notation

Example

The antiderivative of a constant times a function
equals the constant times the antiderivative of the
function.

The antiderivative of the sum (or difference) of two
functions equals the sum (or difference) of their
antiderivatives.

The Fundamental Theorem of Calculus - tells you how to
evaluate definite integrals based on what you know about
antiderivatives.

Rule In English

Rule in Math. Notation

Example

If F(t) is an antiderivative of f(t), then the
definite integral from a to b of f equals the function F
evaluated when t equals b minus F evaluated when t equals
a.